Remark 1.4 Let \(X\)  be a smooth projective variety over a number field \(K\)  with only finitely many \(L\) -points for each number field \(L/K\) . Lang asked whether this finiteness persists over all finitely generated fields of characteristic zero (see [1]}). Lang's philosophy is that the right setting to consider finiteness questions for rational points is that of varieties over finitely generated fields of characteristic zero; see the “Persistence Conjecture” for a precise conjectural statement (e.g., [2]}). Similarly, the right setting to consider density questions for rational points (such as the Hilbert property) is that of varieties over finitely generated fields of characteristic zero, as we pursue here.

